Ab initio structure determination of quasicrystals by density modification method

Journal of Alloys and Compounds 342 (2002) 72–76

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Ab initio structure determination of quasicrystals by density modification method a, a b c c H. Takakura *, A. Yamamoto , M. Shiono , T.J. Sato , A.P. Tsai a

Advanced Materials Laboratory, National Institute for Materials Science, Tsukuba 305 -0044, Japan b Department of Physics, Faculty of Science, Kyushu University, Fukuoka 812 -8581, Japan c Materials Engineering Laboratory, National Institute for Materials Science, Tsukuba 305 -0047, Japan

Abstract A novel density modification method has been applied to phase reconstruction of X-ray single crystal data of quasicrystals. This is based on the low electron density elimination (LDE) method that is an ab initio structure determination method for crystal in the real space. The location, size and shape of the occupation domains for quasicrystals in the n-dimensional (nD) unit cell are obtained without any model structure. Therefore, the LDE method can be a substitute for the direct method for quasicrystals as nD crystals. The structure solution can easily be found in the trial sets (normally 100 sets) and is confirmed by the subsequent analysis. The result is used as a crude starting model for constructing a detailed structure model of quasicrystals. The solutions of several quasicrystals (i-AlPdMn, i-ZnMgHo and i-CdYb) are exemplified. The reliability and limitations of the algorithm to retrieve the quasicrystalline structures will also be discussed.  2002 Elsevier Science B.V. All rights reserved. Keywords: Quasicrystals; Crystal structure and symmetry; X-ray diffraction

1. Introduction Several technical methods for reconstructing structural phases of quasicrystals have been proposed, so far [1–4]. One approach focuses on the use of the Patterson function, together in some cases with contrast variation and the internal (or perpendicular) space dependence of the structure factors [1–3]. This takes advantage of the smooth dependence of the structure factors when plotted with the internal space component of their diffraction vectors. Other approaches find phases using known approximant crystal structures by expressing them as rational cuts of a higherdimensional structure [4]. On the other hand, a standard method for structure solution of crystals, the reciprocal space direct method, has been applied to a model structure of icosahedral quasicrystals [5]. It was demonstrated that the conventional direct method could determine the positions of occupation domains (ODs)1 by assuming their sizes. Elser suggested an ab initio approach for solving quasicrystal structures based on combinatorial optimization programming [6,7]. *Corresponding author. 1 Sometimes other nomenclature, ‘atomic surface’ or ‘window’, is used to describe the same object.

Recently, another ab initio method based on the low density elimination (LDE) method has been applied to real quasicrystals [8]. The LDE method and the Elser’s method give a very similar effect on electron density. Both methods tend to minimize negative density and raise positive peaks. However, the former does not intend to present a complete structure solution. It rather intends to present location of ODs and a rough estimate of their shape. This information is adequate to describe the structure in the higher-dimensional space without an a priori model. Moreover, if a quasicrystal contains heavy atoms as a constituent element, a prominent high-density portion in the density directly suggests which OD contains the heavy atoms. This is quite useful for the early stage of the structure analysis of quasicrytals to obtain an insight of their structures. Therefore, the LDE method would give a short pathway to an atomic model. In the present paper, we describe structure solutions obtained by the LDE method for a model quasicrystal (i-AlPdMn) and several real quasicrystals: i-AlPdMn, i-ZnMgHo and i-CdYb.

2. Method In this part, we describe the procedure of the LDE

0925-8388 / 02 / $ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII: S0925-8388( 02 )00140-8

H. Takakura et al. / Journal of Alloys and Compounds 342 (2002) 72 – 76

method in some detail. The principle of this method is quite simple [9]. The LDE method follows the normal flow of density modification procedure [10]. First we calculate electron density with initial phases and then modify the density with the function,

H

F S D GJ

1 r (r) r 9(r) 5 r (r) 1 2 exp 2 ] ]] 2 0.2rc r 9(r) 5 0

2

r (r) $ 0,

(1)

r (r) , 0,

where r (r) or r 9(r) is the density at position r in the higher-dimensional unit cell in the previous or present cycle, respectively. The rc is the expected average density in the unit cell. From the second cycle a weighted density is calculated. The weight used here is w 5 tanh[uFFc u / hkFlkFc lj],

(2)

where k l denotes the average over all reflections. The weighted density is modified by the function (1) and then Fourier-transformed to give calculated structure factors uFc u, and phases wc . This procedure is repeated until phases are converged, i.e. until the average change of phases becomes less than some predetermined limit (0.58 in the normal case). If convergence is attained, the weight is set to be one. Then the density is modified again by the function (1) for several cycles to obtain the final density. It is important to note that a multi-solution strategy is employed and initial phases are completely random in the LDE method [8]. Here, one might ask how we can extract solutions from resultant densities. The structures of quasicrystals in the higher-dimensional space are simple. Therefore, it is easy to pick up the solutions by investigating all the resultant densities. However, it is desirable to define some figure of merit for evaluating the results, since it enables to minimize one’s effort. For this purpose, a ratio of minimum and maximum density rmin /rmax , is available as will be mentioned in the next section.

1 1 1 1 S5] 2 1 1 1

1

1 1 1 21 21 1

1 1 1 1 21 21

1 21 1 1 1 21

73

1 21 21 1 1 1

1 1 21 . 21 1 1

2

(3)

A primitive cell is employed for structural description of F-type icosahedral quasicrystals. In this case, even- and odd-parity cells are not equivalent to each other.

3.1. Model structure of i-AlPdMn First, the LDE method is applied to determine the phases for the model structure of i-AlPdMn, which has the ] superspace group Fm35 from Yamamoto et al. [12], that is based on the polyhedral OD modeling. The overall feature of this model is similar to that proposed by Boudard et al. [13], which is based on the simple spherical shell modeling. The data set used here has 659 symmetrically independent reflections. The phases are 0 or p, since there is the center of symmetry. Fig. 1 shows the electron density obtained by Fourier synthesis using the correct phases. The three large ODs centered at high symmetry positions (eV:[000000], eB:[111111] / 4 and oB:[311111] / 4) characterize this 6D structure 2 . These ODs correspond to those located at even-parity lattice point, odd-parity lattice point and odd-parity body center (in this order) in Ref. [12]. The OD at eV is composed of Mn core surrounded by Al. Therefore, a peak associated with a relatively wide tail is observed. Although the size of OD at eB is the smallest, it shows the highest density since it contains mainly Pd. The OD at oB shows a peculiar shape, i.e. it shows twin peaks in this cross section. This is due to the fact that the OD is composed of approximately three shells: Mn occupies the core, Pd

3. Structure solutions Here we apply the LDE method to data sets of a model structure of i-AlPdMn and real structures of i-AlPdMn, i-ZnMgHo and i-CdYb. The model structure is useful to describe the power of the present method. Therefore, the results of i-AlPdMn model will be described in detail. Actual calculations of densities were done on a (14)6 -grid in the unit cell by using the Fast Fourier Transform. In order to use reflection data effectively, the similarity transformations [11] were applied to reflection indices, h9 5 (S 21 )n h, n 5 2 for F-type and n 5 3 for P-type quasicrystals, where the transformation matrix S is defined as

Fig. 1. Electron density of the model structure of i-AlPdMn in the fivefold plane of the 6D unit cell. An outermost rectangle with solid lines shows a face-centered unit cell. Several positions are labeled with symbols (eV:[000000], oV:[100000] / 2, eB:[111111] / 4, oB:[311111] / 4, eE:[322222] / 4 and oE:[122222] / 4). r i and r ' indicate external and internal space directions, respectively. 2 The positions of ODs are denoted by symbols based on a primitive cell. Prefix e and o mean even-parity and odd-parity cells, and V, B and E abbreviate vertex, body-center and edge-center, respectively.

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Fig. 2. Reconstructed electron density of the model structure of iAlPdMn in the fivefold plane of the 6D unit cell. The meaning of the symbols is the same as Fig. 1.

occupies the second shell and Al occupies the outermost shell [12,13]. Keeping these structural features in mind, next we try to reconstruct the electron density from its Fourier amplitudes by using the LDE method. Fig. 2 shows a resulting density of phase reconstruction in which 472 out of 659 reflections were assigned with correct phases. All characteristics in the original density (Fig. 1) are reproduced in the reconstructed one. Especially, the diameters of the ODs along the internal space are almost the same as those seen in the original density. It is noticed that there is a small difference between the original and reconstructed densities due to the missed assigned phases that is prominent in the OD centered at oB. Nevertheless, the twin peaks are still observable for this OD. Therefore, one can recognize that there is some kind of shell structure due to heavy elements concerned with this OD, even if the original density is unknown. Fig. 3 shows the plot of rmin /rmax values that were obtained as the result of 100 trial sets of the LDE method.

Fig. 3. Plot of the rmin /rmax values of 100 trial sets for the model structure of i-AlPdMn. The rmin /rmax values were displayed as vertical bars and sorted so as to descend from left to right.

The rmin /rmax value is considered as a simple figure of merit of the LDE method when it is applied to the quasicrystal structures. The left hand side of the plot, small negative rmin /rmax values, indicates the most plausible solutions; the reconstructed densities consist of large positive peaks associated with overall small negative density. On the other hand, the densities corresponding to the right hand side show strong negative peaks. Therefore, they give large negative rmin /rmax values and cannot be the solutions. Typically, if the resulting densities have rmin / rmax values that are greater than 20.1, they show almost the same characteristic feature and can be considered as structure solutions.

3.2. Application on i-AlPdMn We applied the LDE method to an experimental data set of the icosahedral Al 72 Pd 20 Mn 8 [12]. We used 695 symmetrically independent reflections. The space group is ] ˚ In this Fm35 and the 6D lattice parameter is 12.902 A. case, 23 solutions having the same rmin /rmax 5 20.0362 out of 100 trials were obtained. Fig. 4 shows the resulting density. All characteristic features in the density are very similar to those obtained by the model structure of the icosahedral AlPdMn.

3.3. Application on i-ZnMgHo Next, the method has been applied to an experimental data set of an icosahedral Zn 60 Mg 31 Ho 9 [14]. This is the first example in which an unknown quasicrystal structure is solved by an ab initio method [8]. The space group is ] ˚ We used Fm35 and the 6D lattice parameter is 14.62 A. 238 symmetrically independent reflections. Twenty-eight solutions having the same rmin /rmax value of 20.070 out of 100 trials were obtained. The resulting density is shown in Fig. 5, which is the same as that seen in Fig. 2 of Ref. [8], except for the order of similarity transformation. One can see that the structure is very different from that of i-AlPdMn e.g. in Fig. 3 at a glance, although both structures belong to the same superspace group. The structure of i-ZnMgHo is characterized by four ODs

Fig. 4. Reconstructed electron density of i-AlPdMn in the fivefold plane of the 6D unit cell. The meaning of the symbols is the same as Fig. 1.

H. Takakura et al. / Journal of Alloys and Compounds 342 (2002) 72 – 76

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Fig. 5. Reconstructed electron density of i-ZnMgHo in the fivefold plane of the 6D unit cell. The arrow shows an OD located on the low symmetry position (see text). The meaning of the symbols is the same as Fig. 1.

centered at high symmetry positions and an OD located at low symmetry position (indicated with arrow). It seems that the last OD belongs to the OD centered at oV or eB. This quasicrystal might be the most complicated structure ever known. Although it is difficult to discuss a detailed structure of i-ZnMgHo from only this density, one can immediately see that the OD centered at oV contains Ho, since Ho has the large atomic number (Z 5 67) comparing Zn or Mg. By postulating an appropriate size of OD for Ho, an arrangement of Ho atoms can be constructed. It has been specified ˚ and 8.80 A ˚ as the first that a Ho arrangement shows 5.44 A and third nearest neighbor distances as dominant length [8]. Recently, such a Ho arrangement in the i-ZnMgHo has been confirmed by atomic-resolution Z-contrast scanning transmission electron microscopy [15].

3.4. Application on i-CdYb Finally, the method has been applied to an icosahedral Cd 84 Yb 16 that is a stable binary quasicrystal recently found ] [16]. The space group is r m35 and the 6D lattice parame˚ We used 218 symmetrically independent ter is 8.034 A. reflections. Forty-five solutions, having rmin /rmax greater than 20.0568, out of 100 trials were obtained. Fig. 6 shows the resulting density of i-CdYb. The three ODs centered at V:[000000], B:[111111] / 2 and E:[100000] / 2

Fig. 6. Reconstructed electron density of i-CdYb in the fivefold plane of the 6D unit cell. The rectangle with solid lines shows a primitive unit cell. The labels V, B, and E denote high-symmetry positions [000000], [111111] / 2 and [100000] / 2, respectively.

Fig. 7. 2D cut of the 3D electron density of i-CdYb in the external space ˚ ˚ The solid line indicates the perpendicular to a twofold axis (44 A344 A). corresponding edge length of the 3DPT. Note that the atom arrangement in the dashed square is the same as that seen in the rhombic dodecahedron on a symmetry plane perpendicular to the long diagonal in Cd 6 Yb [17].

characterize this 6D structure. The large OD at V suggests that this OD contains heavy Yb. From the analysis of the ‘1 / 1’ approximant structure Cd 6 Yb, it has been suggested that Yb atoms in i-CdYb are located only at the vertices of the three-dimensional ˚ Penrose tiling (3DPT) having an edge length of 5.68 A [17]. It has also been pointed out that there is no atom located on the edge of the 3DPT. Indeed, by taking an irrational cut of the resulting 6D structure as shown in Fig. 7, we see a very similar atom arrangement that exists in the Cd 6 Yb crystal.

4. Discussions As described in the previous section, the LDE method can reconstruct structural phases of quasicrystals when they are described as crystals in the higher-dimensional space. From the reconstructed density, we can directly see the positions of ODs and their rough sizes. This is the clear advantage of using the LDE method compared with e.g. the Patterson based method [1–3]. In our experience, structure solutions were always obtained for quasicrystals. One of the reasons is due to the fact that the structures of quasicrystals are very simple in the higher dimension as displayed in the previous section. All ODs are located at high-symmetry positions in the unit cell. Although we can obtain much information from the resulting densities, some details of the shape of ODs are

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lost. This is apparent from the comparison between Fig. 1 and Fig. 2. In this case, the obscured shapes of the ODs in Fig. 2 are attributed to the missed assignment of phases. However, the original density (Fig. 1) is already suffering from truncation effects. Therefore, there are two factors that smear out the shape of ODs. The density of the icosahedral Al 72 Pd 20 Mn 8 (Fig. 4) is very similar to that obtained from the model structure (Fig. 2). Recently, Brown et al. have also obtained almost the same quality of density for the i-AlPdMn [7]. It has been discussed that i-AlPdMn has much randomness more than that expected so far [7]. Here we only suggest that one needs to be careful to interpret the reconstructed density. Definitive conclusions will be obtained by further analysis by the refinement of an appropriate structure model. Combination with the maximum entropy method [18] will be promising to obtain a more detailed shape of ODs, however. In this case one needs absolute scaled structure factors, which can only be obtained at the final stage of structure refinement. Therefore, next step of the analysis is to make a structure model by using all other available information. The reflection data used in the present study are relatively low resolution data, both in the external and internal spaces. It is interesting to apply the LDE method to more high-resolution data that is obtained by e.g. synchrotron radiation. The present method is also applicable to neutron scattering data, if the scattering lengths of the constituent elements have all positive values.

5. Conclusions The LDE method has been successfully applied to quasicrystals and the solutions for them have been exemplified using both model and experimental data sets. The position of ODs and their rough sizes could be obtained directly from diffraction data. The advantages and limitations of the present method have been depicted. If a quasicrystal contains heavy atoms as a constituent element, the corresponding OD can be specified. On the other hand,

since the shape of OD is smeared out by the truncation effect and the missed assignment of phases, it is difficult to discuss its detailed atomic structure without further knowledge. Finally, by taking a ratio of maximum and minimum density, structure solutions can be easily extracted from the resultant densities without surveying all the results.

Acknowledgements This work is supported by Core Research for Evolutional Science and Technology, Japan Science and Technology Corporation.

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